DOCSLIB.ORG

Composite and Background Fields in Non-Abelian Gauge Models

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Composite and Background Fields in Non-Abelian Gauge Models S S symmetry Article Composite and Background Fields in Non-Abelian Gauge Models Pavel Yu. Moshin 1,* and Alexander A. Reshetnyak 1,2,3 1 Department of Physics, National Research Tomsk State University, 634050 Tomsk, Russia; [email protected] 2 Center for Theoretical Physics, Tomsk State Pedagogical University, 634061 Tomsk, Russia 3 Institute of Strength Physics and Materials Science Siberian Branch of Russian Academy of Sciences, 634021 Tomsk, Russia * Correspondence: [email protected] Received: 19 October 2020; Accepted: 25 November 2020; Published: 30 November 2020 Abstract: A joint introduction of composite and background fields into non-Abelian quantum gauge theories is suggested based on the symmetries of the generating functional of Green’s functions, with the systematic analysis focused on quantum Yang–Mills theories, including the properties of the generating functional of vertex Green’s functions (effective action). For the effective action in such theories, gauge dependence is found in terms of a nilpotent operator with composite and background fields, and on-shell independence from gauge fixing is established. The basic concept of a joint introduction of composite and background fields into non-Abelian gauge theories is extended to the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion, as well as to the Gribov–Zwanziger theory. Keywords: composite fields; background fields; Yang–Mills theory; Volovich–Katanaev model; Gribov–Zwanziger model; background effective action; gauge-dependence 1. Introduction Composite [1,2] and background [3–5] fields are widely used in quantum gauge theories. The attention to composite fields (see [2] for an overview) stems from the fact that the effective action for composite fields suggested in [1] has been applied to quantum field models such as [6–8], including the Early Universe, Inflationary Universe, Standard Model and SUSY theories [9–13]. The study of BRST-invariant renormalizability in Yang–Mills theories, which includes N = 1 SUSY formulations [14–16], the functional renormalization group [17–21] and the Gribov horizon [22–24], appears to be promising within the concept of soft BRST symmetry breaking [25–28] and the local composite operator technique [29,30] as applied to arbitrary backgrounds [31]. In its turn, the background field method [3–5] presents the quantization of Yang–Mills theories using background gauges [5,32,33] in such a way that provides an invariance of the effective action under the gauge transformations of background fields and reproduces physical results with essential simplifications in the Feynman diagrams, thereby providing insight into diverse quantum properties of gauge theories [34–43]; for recent developments, see [44–48]. The present article is devoted to quantum non-Abelian gauge models with composite and background fields, whose consistent analysis isfocused on Yang–Mills theories quantized using the Faddeev–Popov method [49] in a combined presence of composite and background fields. A joint treatment of Yang–Mills fields Am with composite and background ones requires a systematic consideration of these ingredients as interrelated. We forward the symmetry principle as such a systematic concept. In fact, suppose that a generating functional Z (J, L) of Green’s functions with Symmetry 2020, 12, 1985; doi:10.3390/sym12121985 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1985 2 of 26 A composite fields is given, depending on sources JA for the quantum fields f , as well as on sources m Lm for the composite fields s (f). A question then naturally arises of how one can introduce some background fields Bm in a way that leads to an extended functional Z (B, J, L) reflecting the possible symmetries of Z (J, L). Let us next suppose that a generating functional Z (B, J) of Green’s functions, also with certain symmetries, is given in the background field method, and then the question is how m some composite fields s (f, B) with sources Lm can be introduced for the resulting Z (B, J, L) to inherit the given symmetries. It turns out that these two approaches are equivalent in the following sense. According to the first approach, a generating functional Z (J, L) is given, Z i h i Z (J, L) = df exp S (f) + J fA + L sm (f) , (1) h¯ FP A m corresponding to the Faddeev–Popov action SFP (f) of a Yang–Mills theory with composite fields m s (f). A background field Bm can then be introduced by localizing the inherent global symmetry of Z (J, L) under SU(N) transformations (rotations for JA and tensor transformations for Lm) in such a way that Z (B, J, L) defined as Z (B, J, L) = Z (J, L)j (2) ¶m!Dm(B) is invariant under local SU(N) transformations of the sources JA, Lm, accompanied by gauge transformations of the field Bm with an associated covariant derivative Dm (B), where the precise meaning of ¶m ! Dm (B) in (2) is given by (19) of Section2. The original SFP (f) becomes thereby modified to the Faddeev–Popov action of the background field method, SFP (f, B), which is related to SFP (f) by so-called background and quantum transformations of this method (see Section 2.2). According to the second approach, a generating functional Z (B, J) is constructed using the background field method for Yang–Mills theories, Z i h i Z (B, J) = df exp S (f, B) + J fA , (3) h¯ FP A which implies S (f, B) = S (f)j . FP FP ¶m!Dm(B) m Some composite fields s (f, B) with sources Lm can then be introduced on condition that the resulting generating functional Z i h i Z (B, J, L) = df exp S (f, B) + J fA + L sm (f, B) (4) h¯ FP A m should inherit the symmetry of Z (B, J) under local SU(N) rotations of the sources JA accompanied by gauge transformations of the background field Bm with the covariant derivative Dm (B). This symmetry requirement for Z (B, J, L) is satisfied by a local SU(N) tensor transformation law imposed on sm (f, B) m and is provided by Bm entering the composite fields s (f, B) by means of the covariant derivative Dm (B), which implies sm (f, B) = sm (f)j (5) ¶m!Dm(B) for certain sm (f), and thereby we return to the first approach. In the main part of the article, we implement the first approach as a starting point of our systematic analysis, assuming the composite fields to be local, whereas in the remaining part we show how the first and second approaches can be extended beyond the given assumptions by considering the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion [50] and the Gribov–Zwanziger theory [23,24]. Two-dimensional models of gravity [50–61] and supergravity [62–65] are of interest in view of their close relation to string and superstring theory. Simple two-dimensional models also provide a deeper insight into classical and quantum properties of gravity in higher dimensions, while in some Symmetry 2020, 12, 1985 3 of 26 cases these models are exactly solvable at the classical level. One of the two-dimensional gravity models that has been widely discussed at the classical [66–70] and quantum [71–75] levels is the model [50] suggested in the context of bosonic string theory with dynamical torsion [76] in order to address some problems of string theory. Thus, it has been shown [76], using the path integral approach, that a string with dynamical torsion has no critical dimension. The model [50] presents the most general theory of two-dimensional R2-gravity with independent dynamical torsion leading to second-order equations of motion for the zweibein and Lorentz connection. The model also contains solutions with constant curvature and zero torsion, thereby incorporating some other two-dimensional gravity models [51,52,61] whose actions, as compared to that of [50], do not allow a purely geometric interpretation. Being quantized in the background field method, the model yields a gauge-invariant background effective action [75]. In quantum Yang–Mills theories using differential (e.g., Landau or Feynman) gauges, the non-Abelian nature of the gauge group features the Gribov ambiguity [22] implying a residual gauge-invariance due to Gribov copies, which are removed by means of a Gribov horizon [22] implemented in the Gribov–Zwanziger model [23,24] being a quantum Yang–Mills theory in Landau gauge and including an additive horizon functional in terms of a non-local composite field [77]. The issue of bringing the horizon functional to other gauges has been settled due to the concept of finite field-dependent BRST transformations [26,28,78–84], which allows one to present the horizon functional using different gauges in a way consistent with the gauge-independence of the path integral, based on the Gribov–Zwanziger recipe [23,24] and starting from a BRST-invariant Yang–Mills quantum action in Landau gauge. The horizon functional in covariant Rx gauges has been given by [26,28,82,85] (see also [86,87] for a BRST-invariant horizon) and later extended to the Standard Model in [88,89]. The article is organized as follows. In Section2, a generating functional of Green’s functions with composite and background fields in Yang–Mills theories is introduced, as well as a generating functional of vertex Green’s functions (effective action). Analyzing the dependence of the generating functionals of Green’s functions upon a choice of gauge-fixing, we find a gauge variation of the effective action in terms of a nilpotent operator depending on the composite fields and determine the conditions of on-shell gauge-independence. Besides, the effective action Geff (B, S), with a background field Bm and a set of auxiliary tensor fields Sm associated with sm, is found to exhibit a local symmetry m under the gauge transformations of Bm combined with the local SU(N) transformations of S .
Load more

Recommended publications
  • Gauge Theories of the Strong and Electroweak Interaction
    Gauge Theories of the Strong and Electroweak Interaction By Prof. Dr. rer. nat. Manfred Böhm, Universität Würzburg Dr. rer. nat. Ansgar Denner, Paul Scherrer Institut Villigen Prof. Dr. rer. nat. Hans Joos, DESY Hamburg B. G.Teubner Stuttgart • Leipzig • Wiesbaden Contents 1 Phenomenological basis of gauge theories of strong, elec­ tromagnetic, and weak interactions 1 1.1 Elementary particles and their interactions 1 1.1.1 Leptons and quarks as fundamental constituents of matter . 2 1.1.2 Fundamental interactions 4 1.2 Elements of relativistic quantum field theory 5 1.2.1 Basic concepts of relativistic quantum field theory 6 1.2.2 Lie algebras and Lie groups 18 1.2.3 Conserved currents and charges 24 1.3 The quark model of hadrons 29 1.3.1 Quantum numbers and wave functions of hadrons in the quark model 29 1.3.2 Quark model with colour 35 1.3.3 The concept of quark dynamics—quarkonia 36 1.4 Basics of the electroweak interaction 39 1.4.1 Electroweak interaction of leptons 41 1.4.2 Electroweak interaction of hadrons 47 1.5 The quark-parton model 57 1.5.1 Scaling in deep-inelastic lepton-nucleon scattering 57 1.5.2 The parton model 63 1.5.3 Applications of the simple parton model 68 1.5.4 Universality of the parton model 72 1.6 Higher-order field-theoretical effects in QED 76 1.6.1 QED as a quantum field theory 76 Contents 1.6.2 A test of QED: the magnetic moment of the muon 78 1.7 Towards gauge theories of strong and electroweak interactions 81 References to Chapter 1 82 2 Quantum theory of Yang-Mills fields 85 2.1 Green functions and 5-matrix elements 86 2.1.1 The principles of quantum field theory 86 2.1.2 Green functions 88 2.1.3 5-matrix elements and the LSZ formula .
    [Show full text]
  • Effective Potentials and the Vacuum Structure of Quantum Field Theories
    hep-th/0407084 12 Jul 2004 Effective Potentials and the Vacuum Structure of Quantum Field Theories Kristian D. Kennaway Department of Physics and Astronomy University of Southern California Los Angeles, CA 90089-0484, USA Abstract I review some older work on the effective potentials of quantum field theories, in par- ticular the use of anomalous symmetries to constrain the form of the effective potential, and the background field method for evaluating it perturbatively. Similar techniques have recently been used to great success in studying the effective superpotentials of supersymmetric gauge theories, and one of my motivations is to present some of the older work on non-supersymmetric theories to a new audience. The Gross-Neveu model exhibits the essential features of the techniques. In particular, we see how rewriting the Lagrangian in terms of an appropriate composite background field and performing a perturbative loop expansion gives non-perturbative information about the vacuum of the theory (the fermion condensate). The effective potential for QED in a constant electromagnetic background field strength is derived, and compared to the analogous calculation in non-Abelian Yang-Mills theory. The Yang-Mills effective potential shows that the “perturbative” vacuum of Yang-Mills theory is unstable, and the true vacuum has a non-trivial gauge field background. Finally, I describe how some of the limita- tions seen in the non-supersymmetric theories are removed by supersymmetry, which arXiv:hep-th/0407084v1 12 Jul 2004 allows for exact computation of the effective superpotential in many cases. 1 Introduction Work emerging from string theory over the last few years has led to the computation of the exact low-energy effective superpotential for gauge theories with = 1 supersymmetry and N matter in various representations.
    [Show full text]
  • The Background Field Method: Alternative Way of Deriving The
    HUPD-9408 YNU-HEPTh-94-104 May 1994 THE BACKGROUND FIELD METHOD: Alternative Way of Deriving the Pinch Technique’s Results Shoji Hashimoto∗ , Jiro KODAIRA† and Yoshiaki YASUI ‡ Dept. of Physics, Hiroshima University Higashi-Hiroshima 724, JAPAN Ken SASAKI§ Dept. of Physics, Yokohama National University Yokohama 240, JAPAN Abstract We show that the background field method (BFM) is a simple way of deriving the same gauge-invariant results which are obtained by the pinch technique (PT). For illustration we construct gauge-invariant self- energy and three-point vertices for gluons at one-loop level by BFM and arXiv:hep-ph/9406271v1 10 Jun 1994 demonstrate that we get the same results which were derived via PT. We also calculate the four-gluon vertex in BFM and show that this vertex obeys the same Ward identity that was found with PT. ∗Supported in part by the Monbusho Grant-in-Aid for Scientific Research No. 040011. †Supported in part by the Monbusho Grant-in-Aid for Scientific Research No. C-05640351. ‡Supported in part by the Monbusho Grant-in-Aid for Scientific Research No. 050076. §e-mail address: [email protected] 1 Introduction Formulation of a gauge theory begins with a gauge invariant Lagrangian. However, except for lattice gauge theory, when we quantize the theory in the continuum we are under compulsion to fix a gauge. Consequently, the corresponding Green’s functions, in general, will not be gauge invariant. These Green’s functions in the standard formulation do not directly reflect the underlying gauge invariance of the theory but rather obey complicated Ward identities.
    [Show full text]
  • Pion Magnetic Polarisability Using the Background Field Method
    Physics Letters B 811 (2020) 135853 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Pion magnetic polarisability using the background field method ∗ Ryan Bignell , Waseem Kamleh, Derek Leinweber Special Research Centre for the Subatomic Structure of Matter (CSSM), Department of Physics, University of Adelaide, Adelaide, South Australia 5005, Australia a r t i c l e i n f o a b s t r a c t Article history: The magnetic polarisability is a fundamental property of hadrons, which provides insight into their Received 22 May 2020 structure in the low-energy regime. The pion magnetic polarisability is calculated using lattice QCD in the Received in revised form 7 October 2020 presence of background magnetic fields. The results presented are facilitated by the introduction of a new Accepted 8 October 2020 magnetic-field dependent quark-propagator eigenmode projector and the use of the background-field Available online 15 October 2020 corrected clover fermion action. The magnetic polarisabilities are calculated in a relativistic formalism, Editor: W. Haxton and the excellent signal-to-noise property of pion correlation functions facilitates precise values. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction and a hadronic Landau-level projection for the charged pion. The background-field method has previously been used to extract the The electromagnetic polarisabilities of hadrons are of funda- polarisabilities of baryons [18,19]and nuclei [20]in dynamical mental importance in the low-energy regime of quantum chromo- QCD simulations as well as the magnetic polarisability of light dynamics where they provide novel insight into the response of mesons in quenched SU(3) simulations [21–23].
    [Show full text]
  • UNIVERSIDADE ESTADUAL DE CAMPINAS Instituto De F´Isica Gleb Wataghin
    UNIVERSIDADE ESTADUAL DE CAMPINAS Instituto de F´ısica Gleb Wataghin CLARA TEIXEIRA FIGUEIREDO ASPECTS OF DYNAMICAL MASS GENERATION WITHIN THE FORMALISM OF SCHWINGER-DYSON EQUATIONS Aspectos da gera¸c~ao de massa din^amica dentro do formalismo das equa¸c~oesde Schwinger-Dyson CAMPINAS 2020 Clara Teixeira Figueiredo Aspects of dynamical mass generation within the formalism of Schwinger-Dyson equations Aspectos da gera¸c~ao de massa din^amica dentro do formalismo das equa¸c~oes de Schwinger-Dyson Tese apresentada ao Instituto de F´ısica \Gleb Wataghin" da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obten¸c~ao do t´ıtulo de Doutora em Ci^encias, na ´area de F´ısica. Thesis presented to the \Gleb Wataghin" Institute of Physics of the University of Campinas in par- tial fulfillment of the requirements for the degree of Doctor of Science, in the area of Physics. Orientador: Arlene Cristina Aguilar Este exemplar corresponde a` versao~ fi- nal da tese defendida pela aluna Clara Teixeira Figueiredo e orientada pela Profa. Dra. Arlene Cristina Aguilar. Campinas 2020 Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174 Figueiredo, Clara Teixeira, 1991- F469a FigAspects of dynamical mass generation within the formalism of Schwinger- Dyson equations / Clara Teixeira Figueiredo. – Campinas, SP : [s.n.], 2020. FigOrientador: Arlene Cristina Aguilar. FigTese (doutorado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin. Fig1. QCD não perturbativa. 2. Schwinger-Dyson, Equações de. 3. Geração de massa dinâmica. I. Aguilar, Arlene Cristina, 1977-. II.
    [Show full text]
  • Implementation of General Background Electromagnetic Fields on a Periodic Hypercubic Lattice
    Implementation of general background electromagnetic fields on a periodic hypercubic lattice The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Davoudi, Zohreh, and William Detmold. "Implementation of general background electromagnetic fields on a periodic hypercubic lattice." Phys. Rev. D 92, 074506 (October 2015). © 2015 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.92.074506 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/99362 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 92, 074506 (2015) Implementation of general background electromagnetic fields on a periodic hypercubic lattice † Zohreh Davoudi* and William Detmold Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 12 August 2015; published 19 October 2015) Nonuniform background electromagnetic fields, once implemented in lattice quantum chromodynamics calculations of hadronic systems, provide a means to constrain a large class of electromagnetic properties of hadrons and nuclei, from their higher electromagnetic moments and charge radii to their electromagnetic form factors. We show how nonuniform fields can be constructed on a periodic hypercubic lattice under certain conditions and determine the precise form of the background Uð1Þ gauge links that must be imposed on the quantum chromodynamics gauge-field configurations to maintain periodicity. Once supplemented by a set of quantization conditions on the background-field parameters, this construction guarantees that no nonuniformity occurs in the hadronic correlation functions across the boundary of the lattice.
    [Show full text]
  • Background Field Method and Nonlinear Gauges
    Background field method and nonlinear gauges Breno L. Giacchiniab, Peter M. Lavrovcbd and Ilya L. Shapirobcd (a) Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China (b) Departamento de F´ısica, ICE, Universidade Federal de Juiz de Fora Juiz de Fora, CEP: 36036-330, MG, Brazil (c) Department of Mathematical Analysis, Tomsk State Pedagogical University, 634061, Kievskaya St. 60, Tomsk, Russia (d) National Research Tomsk State University, Lenin Av. 36, 634050 Tomsk, Russia E-mails: [email protected], [email protected], shapiro@fisica.ufjf.br Abstract We present a reformulation of the background field method for Yang-Mills type theories, based on using a superalgebra of generators of BRST and background field transformations. The new approach enables one to implement and consistently use non-linear gauges in a natural way, by using the requirement of invariance of the fermion gauge-fixing functional under the background field transformations. Keywords: background field method, nonlinear gauge, Yang-Mills theories, quantum gravity theories 1 Introduction arXiv:1906.04767v2 [hep-th] 10 Sep 2019 The standard approach to the Lagrangian quantization of gauge theories [1–3] assumes the violation of the gauge invariance of the action. In some cases, e.g., in the semiclassi- cal gravity theory [4], this makes all considerations and also practical calculations quite complicated. The same certainly concerns quantum gravity. Fortunately, there is a useful approach to the quantization of gauge theories, known as the background field method (BFM) [5–7]. Within the BFM one can explicitly preserve the gauge invariance of an ef- fective action in all stages, and thus all physical results are reproduced using the effective action of background fields (background effective action).
    [Show full text]
  • Electromagnetic Polarizabilities from Lattice QCD
    Electromagnetic Polarizabilities from Lattice QCD W. Detmold (College of William and Mary) * B. Tiburzi (University of Maryland) A. Walker-Loud (College of William and Mary) Outline Physics Motivation: Low-energy hadron structure chiral symmetry electromagnetic polarizabilities Lattice QCD: Background field method Results: Electric Polarizabilities Pions Nucleons Physics Motivation Spontaneous chiral symmetry breaking has important consequences on low energy hadron structure In external EM fields, hadrons polarize against strong chromodynamic forces p! = α E! − E Chiral dynamics: charged pion cloud 1 ∆H = α E" 2 deforms (any hadron) −2 E Hadronic polarizabilities are tightly constrained by chiral dynamics 2 2 π e p e αE = N 2 αE = N ! 2 mπΛ mπΛ Electromagnetic Polarizabilities One-loop result + (Holstein) π Two-loop result (Gasser, Ivanov & Sainio) Mainz extraction (Ahrens, et al.) N Compton Scattering Theory: include delta resonance? magnetic isovector? New data 60, 100 MeV Extraction: deuteron ( 3 He plans) ∼ taken! ! 100 MeV 65 MeV Extraction: ∼ approved Lattice QCD Use first principles numerical techniques to + calculate polarizabilities directly from QCD...? Stable hadrons! neutron, pion, etc. Lattice 4-point functions: not enough time! Not enough space! 2πn ψ(x + L)=ψ(x) k = 450 MeV → L ∼ Also limitation for 3-point functions: magnetic moment, quark orbital angular momentum Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action E.g. neutral pion in electric field Aµ = x4 δµ3 −E 1 0 2 2 0 (p! =0,x4)= π ∂4∂4 + mπ0 + mπ0 αE π L 2 ! − E " G(τ)= χ(#x, τ)χ†(#0, 0) = Z exp( Eτ) ! " − !!x 1 2 E = m 0 + α π 2 E E E.g.
    [Show full text]
  • Use of the Heat-Kernel Expansion with the Background Field Method to Regularize Supersymmetric (Gauge) Theories
    Revista Brasileira de Flsica, Vol. 15, n? 1, 1985 Use of the Heat-Kernel Expansion with the Background Field Method to Regularize Supersymmetric (Gauge) Theories E. ABDALLA CERN, Geneva, and Instituto de Física, Universidade de 30Paulo, Caixa Postal 20516, São Paulo, O1000, SP, Brasil and M.C.B. ABDALLA Instituto de Física Tedrica, Caixa Postal 5956, Sáo Paulo, 01000. SP, Brasil Recebido em 4 de junho de 1985 Abstiact We use the proper time variable of the heat-kernel expans ion to regularize field theories in the framework of the background f ield method. The method can be naturally applied to supersymme~ricand gauge theories. Explicit computations have been done including superfield perturbation theory. 1. INTRODUCTION Renormal ization is one of the most difficult techniques in quantum field theory. üesides difficulties of principle with some schemes breaking down at very high orders of perturbation theoryl, some symmetries of the theory can be rnutilated in the process. The problem of recovering gauge symmetry in the BPHZ scheme2 is notorious3. Dimen- sional regularization4 is the best method to deal with gauge theories, since the gauge principle is maintained in arbitrary dimensions. How- ever, concerning supersymmetric theories the situation is not yet in good shape. This comes from the fact that the number of degrees of freedom of fields of different spin does not behave in the same way with respect to the dimension. A way out is the process called dimen- sional regularization via dimensional reduction, or supersymmetric di- mensional regularization (SDR) '. In this process, the space-time is D- dimensional, but the fields and the algebra are kept 4-d imens ional .
    [Show full text]
  • Further Comments on the Background Field Method and Gauge Invariant
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server GEF-TH-04/994 Further comments on the background field method and gauge invariant effective actions. Carlo Becchi∗ Renzo Collina† Dipartimento di Fisica dell' Universit´adiGenova and INFN, Sezione di Genova, Via Dodecaneso 33, I-16146, Genova, Italy ABSTRACT The aim of this paper is to give a firm and clear proof of the existence in the background field framework of a gauge invariant effective action for any gauge theory (background gauge equivalence). Here by effective action we mean a functional whose Legendre transform restricted to the physical shell generates the matrix elements of the connected S-matrix. We resume and clarify a former argument due to Abbott, Grisaru and Schaefer based on the gauge-artifact nature of the background fields and on the identification of the gauge invariant effective action with the generator of the proper, background field, vertices. ∗E-mail: [email protected] †E-mail: [email protected] 1 1. Introduction The analysis at LEP of the effective couplings of the intermediate bosons of the electro-weak interactions has further increased the need of an efficient parametrization method for the low energy effective actions of gauge theories. In general these effective actions are identified with the functionals generating the fully renormalized vertices and propagators contributing to the skeleton graphs, technically speaking, the proper, 1-particle irreducible, amplitudes. Therefore they are controlled by the non-linear Slavnov-Taylor identity accounting for the BRS symmetry of the gauge theories.
    [Show full text]
  • Background Field Method for D=3, N=3 Supersymmetric Gauge
    BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIES I.L. Buchbinder TSPU, Tomsk, Russia The VIII International Workshop "Supersymmtries and Quantum Symmetries", Dubna, July 29 { August 3,2009 I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 1 / 24 Aims Formulation of gauge invariant effective action for d = 3, N = 3 supersymmetric gauge theories, formulated in d = 3, N = 3 harmonic superspace. Construction of manifestly N = 3 supersymmetric effective action in terms of off-shell d = 3, N = 3 superfields. d = 3, N = 3 harmonic supergraph calculations. Talk is based on the results obtained in collaboration with E. Ivanov, O. Lechtenfeld, N. Pletnev, I. Samsonov, B. Zupnik I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 2 / 24 Motivations Studing a quantum structure of recently proposed Bagger-Lambert-Gustavsson (BLG) theory (J. Bagger, N. Lambert, Phys. Rev. D75 (2007) 045020; D77 (2008) 065008; JHEP 0802 (2008) 105; A. Gustavsson, JHEP 0804 (2008) 083) and Aharony-Bergman-Jefferis-Maldacena (ABJM) theory (O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, JHEP 0810 (2008) 091). Both theories related to low-energy description of M2-branes and contain the Chern-Simons fields and some number of scalar and spinor fields. These theories possess N = 8 supersymmetry N = 6 supersymmetry resepectively. One can show that BLG and ABJM theories can be formulated in terms of d = 3, N = 3 harmonic superspace (I.L.B, E.A. Ivanov, O. Lechtenfeld, N.G. Pletnev, I.B. Samsonov, B.M. Zupnik, JHEP 03 (2009) 096).
    [Show full text]
  • Application of the Functional RG with Background Field Methods
    Application Of The Functional RG With Background Field Methods J. Braun Heidelberg University December 9, 2005 TU Darmstadt Outline Motivation Functional Renormalization Group - General Introduction Gauge Theories And Functional Renormalization Group Applications: QCD at finite temperature Conclusions and outlook Motivation - challenging questions of QCD ◮ running QCD coupling −→ small momentum behavior? ◮ confinement −→ mechanisms? ◮ chiral symmetry breaking −→ Tχ = Td ? ◮ many-flavor QCD ◮ phase diagram of QCD (Bethke ’04) −→ order of the phase transition(s)? ◮ QCD at high temperature ◮ topological effects in QCD −→ e.g. role of instantons? (Karsch et al. ’03) Motivation - Why Renormalization Group? Lattice QCD ◮ implementation of fermions (staggered, Wilson, domain wall, ...) is difficult ◮ large current quark masses −→ extrapolation required (chiral perturbation theory) ◮ finite volume −→ IR-cutoff introduced −→ extrapolation required ◮ chemical potential −→ sign problem Why Renormalization Group? ◮ allows to describe physics across different length scales (microscopic theory → macroscopic theory) ◮ allows for chiral fermions ◮ implementation of quarks with and without current quark masses ◮ critical behavior: importance of long-range fluctuations ◮ study QFT’s in infinite and finite volume ◮ non-perturbative method ◮ truncation is required =⇒ complementary method to Lattice QCD Motivation - Why Renormalization Group? Lattice QCD ◮ implementation of fermions (staggered, Wilson, domain wall, ...) is difficult ◮ large current quark masses −→ extrapolation
    [Show full text]